Timeline for Situations where “naturally occurring” mathematical objects behave very differently from “typical” ones
Current License: CC BY-SA 4.0
16 events
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| Aug 16, 2021 at 14:15 | history | edited | Theodore Norvell | CC BY-SA 4.0 |
Clarified why this answer does not fit all the criteria asked for in the the question.
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| Aug 11, 2021 at 19:42 | comment | added | Theodore Norvell | @MattF. You are right, there is a clear (to most mathematicians and computer scientists) explanation. I focused on scarcity, and missed the requirement for mystery. I'm am down-voting this answer. (Darn, it wouldn't let me.) | |
| Aug 11, 2021 at 19:38 | comment | added | Theodore Norvell | @darijgrinberg You might be right. I don't see it. I think this is a another level of scarcity in the following sense. Given a random real number, the probability that it computable is is 0 in the sense that it is |N| out of |R|. If you compare the set of all provable conjectures to the set of all true conjectures, you get |N| out of |N|. Given a random true conjecture, there is a finite probability that it is provable. What is 0 is the the limit of that probability as a length limit increases: lim n->inf :: |{ t in T | #t < n and t is provable}|/{ t in T #t < n}|, where # is length. | |
| Aug 11, 2021 at 19:12 | comment | added | user44143 | @TheodoreNorvell, I can restate my comment without that word: “Often we develop new mathematics in order to compute more. This basic observation provides a clear explanation for the phenomenon that the numbers occurring in our mathematics are mostly computable numbers. As a result, the example here does not answer the question.” | |
| Aug 11, 2021 at 19:03 | comment | added | Theodore Norvell | @MattF No it's not surprising that mathematician mostly use computable numbers. I don't think the OP required the result to be surprising. Although, back in the 19th c. a lot of people were surprised to learn that that most numbers were incomputable (although they didn't use that term). I'm still entirely comfortable with the idea. | |
| Jul 28, 2021 at 20:25 | comment | added | darij grinberg | Yeah, in hindsight I should have rather said "This is not very surprising given Timothy Chow's answer". | |
| Jul 28, 2021 at 20:15 | comment | added | JoshuaZ | @darijgrinberg That's roughly true, but that doesn't say much about the distribution of the uncomputable reals. | |
| Jul 28, 2021 at 20:10 | comment | added | darij grinberg | @JoshuaZ: I'm not an expert in the subject and will likely make a mess of it, but let me try: Deciding a sufficiently general(?) kind of statements is equivalent(?) to deciding the halting of a Turing machine, but this can in turn be encoded as computing a real number. I believe Chaitin popularized these equivalences. | |
| Jul 28, 2021 at 17:16 | comment | added | JoshuaZ | @darijgrinberg Can you expand on how they are equivalent? This isn't obvious to me. | |
| Jul 28, 2021 at 17:06 | comment | added | user44143 | I see a clear explanation for this phenomenon: We are interested in computing numbers! Often we develop new mathematics in order to compute more. So it is no surprise that the numbers occurring in our mathematics are mostly computable numbers. | |
| Jul 28, 2021 at 16:22 | comment | added | Captain Giraffe | @darijgrinberg Me coming from the HNQ find this a lot more tangible as a layman. Your comment made me a lot more interested in Timothys answer. | |
| Jul 28, 2021 at 12:03 | comment | added | darij grinberg | This is equivalent to Timothy Chow's answer :) | |
| Jul 28, 2021 at 9:12 | history | edited | John Bentin | CC BY-SA 4.0 |
punctuation, grammar
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| Jul 27, 2021 at 20:46 | review | First posts | |||
| Jul 27, 2021 at 20:55 | |||||
| S Jul 27, 2021 at 20:45 | history | answered | Theodore Norvell | CC BY-SA 4.0 | |
| S Jul 27, 2021 at 20:45 | history | made wiki | Post Made Community Wiki by Theodore Norvell |